The binding energy of a H-atom, considering an electron moving around a fixed nuclei (proton), is
(m = electron mass).
If one decides to work in a frame of reference where the electron is at rest, the proton would be moving around it. By similar arguments, the binding energy would be
This last expression is not correct because
(a) n would not be integral.
(b) Bohr-quantisation applies only to electron
(c) the frame in which the electron is at rest is not inertial.
(d) the motion of the proton would not be in circular orbits, even approximately.